Onsager coefficients and transference numbers

The idea

Conductivity (Lesson 6.5) tells you how much current flows. In a battery you also want to know which ions carry it: ideally Li⁺, because anion current only builds up concentration gradients. The transference number t₊ is the fraction of the current carried by the cations.

Onsager coefficients. For each species i, add up the displacements of all its ions: ΔR_i(t) = Σ Δr over ions of i. The coefficients

L^ij = (1/6VkT) · d/dt ⟨ΔR_i(t) · ΔR_j(t)⟩

describe how strongly species i and j move together. For a 1:1 salt there are three: L⁺⁺, L⁻⁻ and L⁺⁻. They connect forces (gradients of electrochemical potential) to fluxes, like a matrix version of Fick's law. Each diagonal Lⁱⁱ splits into a self part (each ion's own MSD, i.e. N_iD_i) and a distinct part (different ions of the same species moving together); L⁺⁻ is purely distinct.

Conductivity and transference follow directly (z_± = ±1):

σ = e²(L⁺⁺ + L⁻⁻ − 2L⁺⁻),    t₊ = (L⁺⁺ − L⁺⁻) / (L⁺⁺ + L⁻⁻ − 2L⁺⁻)

What clusters do (code tab, Part 1):

• Neutral pairs (LiA) raise L⁺⁻: they lower σ but leave t₊ almost unchanged, because they carry no charge.
• Negative clusters such as [LiA₂]⁻ carry Li⁺ towards the positive electrode. If they are mobile and free Li⁺ is slow, t₊ can become negative, as reported for some concentrated electrolytes and polymer electrolytes.
• The "apparent" t₊ from self-diffusion, D₊/(D₊ + D₋) (what pulsed-field-gradient NMR gives), ignores all cross terms. Here it stays around 0.45 while the true t₊ is −0.24.

The reference frame matters. Fluxes must be measured relative to something: the solvent (solvent-fixed frame), the centre of mass (barycentric frame, what MD naturally conserves) or the centre of volume. In a pure binary molten salt or ionic liquid with no solvent, momentum conservation forces m₊ΔR₊ + m₋ΔR₋ = 0. Then all three L's are tied together and the barycentric t₊ = m₋/(m₊ + m₋), whatever the dynamics (Part 2). Experimental transference numbers are usually defined relative to the solvent or the anion; convert the frame before you compare.

The algorithm

save UNWRAPPED positions; subtract the reference-frame motion (e.g. solvent centre of mass, or total COM)
R_i(t) = Σ_{ions of species i} u(t)                   # one vector per species per frame
for each pair (i, j) and lag k:
    C_ij[k] = < (R_i(t0+k) - R_i(t0)) · (R_j(t0+k) - R_j(t0)) >_t0
L_ij = slope of C_ij in the linear region / (6 V kB T)
self part of L_ii = N_i D_i / (V kB T);   distinct part = L_ii - self
σ   = e² Σ_ij z_i z_j L_ij
t_+ = (z_+² L_++ + z_+ z_- L_+-) · e² / σ

1. Statistics are collective (one R per species), so the same advice as for σ applies: long runs, several replicas, error bars. Off-diagonal L's are the noisiest.
2. For an electrolyte with solvent, work in the solvent-fixed frame to compare with Hittorf or concentration-cell experiments; Bruce–Vincent (polarization) measurements give a related but different quantity in concentrated systems.
3. Check: L⁺⁻ > 0 means cations and anions move together (pairing); negative values mean they move oppositely (as in the barycentric frame of a molten salt).
4. Further reading: Wheeler and Newman (2004) for the concentrated-solution framework and reference frames; Fong et al. (2020, 2021) for Onsager coefficients from MD of polymer electrolytes. Most groups compute the L's with their own scripts, as here.